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A284839 Number of compositions (ordered partitions) of n into prime power divisors of n (including 1).

%I A284839 #7 Feb 16 2025 08:33:43
%S A284839 1,1,2,2,6,2,24,2,56,20,128,2,1490,2,741,449,5272,2,36901,2,81841,
%T A284839 3320,29966,2,4135004,572,200389,26426,5452795,2,110187694,2,47350056,
%U A284839 226019,9262156,51885,10783889706,2,63346597,2044894,14064551462,2,109570982403,2,35537376325,470326038,2972038874,2
%N A284839 Number of compositions (ordered partitions) of n into prime power divisors of n (including 1).
%H A284839 Alois P. Heinz, <a href="/A284839/b284839.txt">Table of n, a(n) for n = 0..1000</a>
%H A284839 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePower.html">Prime Power</a>
%H A284839 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A284839 a(n) = [x^n] 1/(1 - x - Sum_{p^k|n, p prime, k>=1} x^(p^k)).
%F A284839 a(n) = 2 if n is a prime.
%e A284839 a(4) = 6 because 4 has 3 divisors {1, 2, 4} and all are prime powers therefore we have [4], [2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2] and [1, 1, 1, 1].
%p A284839 with(numtheory):
%p A284839 a:= proc(n) local d, b; d, b:= select(x->
%p A284839       nops(factorset(x))<2, divisors(n)),
%p A284839       proc(n) option remember; `if`(n=0, 1,
%p A284839         add(`if`(j>n, 0, b(n-j)), j=d))
%p A284839       end: b(n)
%p A284839     end:
%p A284839 seq(a(n), n=0..60);  # _Alois P. Heinz_, Apr 15 2017
%t A284839 Table[d = Divisors[n]; Coefficient[Series[1/(1 - x - Sum[Boole[PrimePowerQ[d[[k]]]] x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 47}]
%Y A284839 Cf. A000961, A066882, A100346, A284465.
%K A284839 nonn
%O A284839 0,3
%A A284839 _Ilya Gutkovskiy_, Apr 03 2017